The congruence subgroup property for the hyperelliptic modular group: the open surface case

2008/03/27 by Boggi, Marco
#11F80 #14F35 #14H10 #14H15 #14H30 #30F60 #32G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0803.3841

Abstract

Let \cal M_g,n and \cal H_g,n, for 2g-2+n>0, be, respectively, the moduli stack of n-pointed, genus g smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with Γ_g,n and H_g,n, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on \cal H_g,n defines a monomorphism H_g,n\hookrightarrowΓ_g,n. Let S_g,n be a compact Riemann surface of genus g with n points removed. The Teichmüller group Γ_g,n is the group of isotopy classes of diffeomorphisms of the surface S_g,n which preserve the orientation and a given order of the punctures. As a subgroup of Γ_g,n, the hyperelliptic modular group then admits a natural faithful representation H_g,n\hookrightarrowOut(π₁(S_g,n)). The congruence subgroup problem for H_g,n asks whether, for any given finite index subgroup H^λ of H_g,n, there exists a finite index characteristic subgroup K of π₁(S_g,n) such that the kernel of the induced representation H_g,n\toOut(π₁(S_g,n)/K) is contained in H^λ. The main result of the paper is an affirmative answer to this question for n≥ 1.

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